The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model
Bibliographic record
Abstract
Asymmetric spike patterns are constructed for the two‐component Schnakenburg reaction–diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields k 1 spikes that have a common small amplitude and k 2 = k − k 1 spikes that have a common large amplitude. A k ‐spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction–diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O (1) time scale.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".